I will prove Gromov's conjecture that every 3-manifold of
positive scalar curvature contains a short closed geodesic. The
proof uses Min-Max theory of minimal surfaces and a combinatorial
version of mean curvature flow. Time permitting, I will...
Ricci solitons, introduced by R. Hamilton in the mid-80s, are
self-similar solutions to the Ricci flow and natural
generalizations of Einstein manifolds. Shrinking Ricci solitons, in
particular, model Type I singularities of the Ricci flow
and...
In 2003, Bressan proposed a conjecture on the mixing efficiency
of incompressible flows, which remains open. This talk surveys
progress toward resolving Bressan’s mixing conjecture and presents
a new result confirming its asymptotic validity for...
Ricci solitons are the self-similar solutions to the Ricci flow,
which is the heat equation for Riemannian metrics, and they model
singularity formation. We survey various estimates for Ricci
solitons in dimension 4. This is mainly the work of...
Remarkable martensitic microstructures are observed in the
alloy Ti76Nb22Al2 , which undergoes a cubic to orthorhombic
transformation with six martensitic variants Ui=UTi greater than 0
having middle eigenvalue λ2(Ui) very close to 1. Assuming
that...
This talk is concerned with solutions of the 3D incompressible
Navier-Stokes equations that are bounded in a critical space. From
small initial data, these solutions are known to be globally
well-posed due to classical work of Fujita-Kato and others...
The compressible Euler equation can lead to the emergence of
shock discontinuities in finite time, notably observed behind
supersonic planes. A very natural way to justify these
singularities involves studying solutions as inviscid limits of
Navier...
The Hopf-Tsuji-Sullivan theorem states that the geodesic flow on
(an infinite) Riemann surface is ergodic iff the Poincare series is
divergent iff the Brownian motion is recurrent. Infinite Riemann
surfaces can be built by gluing infinitely many...
